# Introduction¶

This package provides reconstruction algorithms for diffraction tomography in two and three dimensions.

## Installation¶

To install via the Python Package Index (PyPI), run:

pip install odtbrain

On some systems, the FFTW3 library might have to be installed manually before installing ODTbrain. All other dependencies are installed automatically. If the above command does not work, please refer to the installation instructions at the GitHub repository or create an issue

## Theoretical background¶

A detailed summary of the underlying theory is available in [MSG15b].

The Fourier diffraction theorem states, that the Fourier transform \(\widehat{U}_{\mathrm{B},\phi_0}(\mathbf{k_\mathrm{D}})\) of the scattered field \(u_\mathrm{B}(\mathbf{r_D})\), measured at a certain angle \(\phi_0\), is distributed along a circular arc (2D) or along a semi-spherical surface (3D) in Fourier space, synthesizing the Fourier transform \(\widehat{F}(\mathbf{k})\) of the object function \(f(\mathbf{r})\) [KS01], [Wol69].

In this notation, \(k_\mathrm{m}\) is the wave number, \(\mathbf{s_0}\) is the norm vector pointing at \(\phi_0\), \(M=\sqrt{1-s_\mathrm{x}^2}\) (2D) and \(M=\sqrt{1-s_\mathrm{x}^2-s_\mathrm{y}^2}\) (3D) enforces the spherical constraint, and \(l_\mathrm{D}\) is the distance from the center of the object function \(f(\mathbf{r})\) to the detector plane \(\mathbf{r_D}\).

## Fields of Application¶

The algorithms presented here are based on the (scalar) Helmholtz equation. Furthermore, the Born and Rytov approximations to the scattered wave \(u(\mathbf{r})\) are used to linearize the problem for a straight-forward inversion.

The package is intended for optical diffraction tomography to determine the refractive index of biological cells. Because the Helmholtz equation is only an approximation to the Maxwell equations, describing the propagation of light, FDTD simulations were performed to test the reconstruction algorithms within this package. The algorithms present in this package should also be valid for the following cases, but have not been tested appropriately:

tomographic measurements of absorbing materials (complex refractive index \(n(\mathbf{r})\))

ultrasonic diffraction tomography, which is correctly described by the Helmholtz equation